To prepare broth, combine first 9 ingredients in a Dutch oven; bring to a boil. Cover, reduce heat, and simmer 45 minutes. Strain through a sieve into a bowl; discard solids. Stir in soy sauce.
Preheat oven to 400°.
Melt butter in a large nonstick skillet over medium heat. Add 2 cups mushrooms, 1 cup leek, and 1 cup carrot; cook 10 minutes or until tender, stirring occasionally. Set aside.
To prepare gnocchi, pierce potatoes with a fork; bake at 400° for 1 hour or until tender. Cool slightly. Peel potatoes; mash. Lightly spoon flour into dry measuring cups, and level with a knife. Combine potatoes, 1 1/4 cups flour, 1 teaspoon salt, and egg yolk. Stir well to form a smooth dough. Turn dough out onto a lightly floured surface. Knead until smooth; add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands. Divide dough into 4 equal portions, and shape each portion into a 15-inch-long rope. Cut each rope into 15 (1-inch) pieces, and roll each piece into a ball. Drag a fork's tines over each ball, forming a concave shape. Place the gnocchi on a baking sheet coated with cooking spray. Cover and set aside.
Bring water to a boil in a large Dutch oven. Add half of gnocchi; cook 1 1/2 minutes or until gnocchi float. (Do not overcook or gnocchi will fall apart.) Remove gnocchi with a slotted spoon; place in a colander. Repeat procedure with remaining gnocchi. Bring broth to a simmer; stir in mushroom mixture. Place 10 gnocchi in each of 6 soup bowls. Ladle 1 cup broth mixture into each bowl; sprinkle with chives. Serve immediately.